A simple random sample of 20 pages from a dictionary is obtained. The numbers of words defined on those pages are found, with the results $n=20, \bar{x}=58.1$ words, $s=15.2$ words. Given that this dictionary has 1476 pages with defined words, the claim that there are more than 70,000 defined words is equivalent to the claim that the mean number of words per page is greater than 47.4 words. Use a 0.05 significance level to test the claim that the mean number of words per page is greater than 47.4 words. What does the result suggest about the claim that there are more than 70,000 defined words? Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim. Assume that the population is normally distributed. What are the null and alternative hypotheses? A. $\mathrm{H}_{0}: \mu>47.4$ words $\mathrm{H}_{1}: \mu<47.4$ words C. $\mathrm{H}_{0}: \mu=47.4$ words $\mathrm{H}_{1}: \mu>47.4$ words B. $\mathrm{H}_{0}: \mu=47.4$ words $\mathrm{H}_{1}: \mu<47.4$ words D. $\mathrm{H}_{0}: \mu=47.4$ words $\mathrm{H}_{1}: \mu \neq 47.4$ words Determine the test statistic. (Round to two decimal places as needed.) Determine the P-value. (Round to three decimal places as needed.) State the final conclusion that addresses the original claim. $\mathrm{H}_{0}$. There is evidence to support the claim that there 70,000 defined words in the dictionary.
Solution
Step 1 ：State the null and alternative hypotheses. The null hypothesis is that the mean number of words per page is equal to 47.4 words and the alternative hypothesis is that the mean number of words per page is greater than 47.4 words. So, we have: \[\mathrm{H}_{0}: \mu=47.4 \text{ words} \] \[\mathrm{H}_{1}: \mu>47.4 \text{ words}\]
Step 2 ：Calculate the test statistic using the formula for the test statistic for a sample mean, which is (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size)). Using the given values, we get: \[t = \frac{\bar{x} - \mu_{0}}{s/\sqrt{n}} = \frac{58.1 - 47.4}{15.2/\sqrt{20}} \approx 3.15\]
Step 3 ：Find the P-value. The P-value is the probability of obtaining a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. Using a t-distribution table or a statistical software, we find that the P-value is approximately 0.003.
Step 4 ：Make a conclusion. Since the P-value (0.003) is less than the significance level (0.05), we reject the null hypothesis. There is evidence to support the alternative hypothesis that the mean number of words per page is greater than 47.4 words. This suggests that there are more than 70,000 defined words in the dictionary.
Step 5 ：Final Answer: The null and alternative hypotheses are: \[\mathrm{H}_{0}: \mu=47.4 \text{ words} \] \[\mathrm{H}_{1}: \mu>47.4 \text{ words}\] The test statistic is approximately \(\boxed{3.15}\). The P-value is approximately \(\boxed{0.003}\). Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. There is evidence to support the claim that the mean number of words per page is greater than 47.4 words. This suggests that there are more than 70,000 defined words in the dictionary.
